Here is where I get lost. I've spent hours re-arranging the numerator and denominator through basic arithmetic operations and trying to get the denominator to cancel out in some way but to no avail. I've spent the last couple of days trying to solve this problem in various ways which is quite embarrassing because the problem seems so simple.

I've also trying setting [tex]T_A=T_B[/tex] and setting the equation equal to zero:

Here, from factoring the term "(a-b)" and finding the zeroes, i understand a=b, therefore a-b=0, but by substituting the term (a-b) with 0, i get 0. By substituting all a or b terms with the other, i get x=a thus x=b since a=b.

All right, I finally got this one, I think. Anyways, my solution is probably not the quickest, but I'll provide an outline and try to find a quicker solution later, maybe.

Let a and b be the points we are trying to determine. Assume a and b are distinct. From f'(a) = f'(b), we get an equality relating a and b (and the number 1). Denote this equality as ***

Without loss of generality, assume b > a. Then [f(b)-f(a)]/(b-a) = f'(b) = f'(a). After lots of factoring and simplification, I got [f(b)-f(a)]/(b-a) = f'(b) down to expressions where the only thing I could use was the equality ***. Applying that twice, I got an extremely simple equality. In fact, you should be able to derive *** and even guess what a and b are. But yeah, from that point onwards, there are probably quicker ways to actually find a and b.

Actually, you missed a solution in setting the constants equal. I essentially reached the same equation by a slightly longer route. Note that the assumption a =/= b is an important one to make in this problem. To summarize your steps:

We get

[tex](a-b)(a^2+ab+b^2-1)=0[/tex]

by setting the slopes equal to each other. Assuming that a =/= b gives